Engineering Physics

Rotational dynamics, thermodynamic cycles, and heat engines.

Spec Points Covered
  • Define angular displacementThe angle through which an object rotates, measured in radians. One complete revolution = $2\pi$ rad., angular velocity, and angular acceleration and relate them to their linear equivalents.
  • Apply the rotational kinematic equations: $\omega = \omega_0 + \alpha t$, $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$, $\omega^2 = \omega_0^2 + 2\alpha\theta$.
  • Define torqueThe rotational equivalent of force. $T = Fr\sin\theta$, where $r$ is the perpendicular distance from the axis. Measured in N m. as $T = Fr$ and explain its role as the rotational equivalent of force.
  • Define moment of inertiaThe rotational equivalent of mass. $I = \sum mr^2$. It depends on both the mass and how that mass is distributed relative to the axis. $I = \sum mr^2$ and use given values for standard shapes.
  • Apply Newton's second law for rotation: $T = I\alpha$.
  • Apply the conservation of angular momentum$L = I\omega$. In the absence of external torques, angular momentum is conserved: if $I$ increases, $\omega$ decreases, and vice versa. $L = I\omega$.
  • Define angular impulse $\Delta L = T\Delta t$ and relate it to change in angular momentum.
  • Calculate rotational work $W = T\theta$ and power $P = T\omega$.
  • Calculate rotational kinetic energy $E_k = \frac{1}{2}I\omega^2$ and combine with translational KE for rolling objects.
  • Describe the function of flywheelsHeavy rotating wheels used to smooth engine output. They store rotational kinetic energy during power strokes and release it between strokes. in smoothing engine output and storing rotational energy.
  • State and apply the first law of thermodynamics: $Q = \Delta U + W$.
  • Interpret p-V diagramsGraphs of pressure against volume for a gas. The area under the curve equals the work done by or on the gas. and calculate work done as the area under the curve.
  • Describe isothermal, adiabatic, isobaric, and isochoric processes and their shapes on a p-V diagram.
  • Describe the four-stroke petrol engine cycle (Otto cycleThe idealised thermodynamic cycle for a petrol engine: two adiabatic and two isochoric processes forming a closed loop on a p-V diagram.) and represent it on a p-V diagram.
  • Describe the diesel engine cycle and explain how it differs from the Otto cycle.
  • Compare petrol and diesel engines in terms of compression ratio, efficiency, and power output.
  • Define and calculate input power, indicated power, brake power, and friction power.
  • Define and calculate overall, thermal, and mechanical efficiency.
  • State the second law of thermodynamics and explain why a heat engine needs a source and a sink.
  • Calculate the maximum theoretical (CarnotThe maximum possible efficiency for a heat engine operating between two temperatures: $\eta = 1 - T_C/T_H$. No real engine can reach this.) efficiency $\eta = 1 - T_C / T_H$.
  • Explain why real engines are always less efficient than the Carnot limit.
  • Describe the operation of reversed heat engines (refrigerators and heat pumps).
  • Define and calculate the coefficient of performanceA measure of how effectively a reversed heat engine transfers heat. For a refrigerator: $\text{COP} = T_C / (T_H - T_C)$. For a heat pump: $\text{COP} = T_H / (T_H - T_C)$. for refrigerators and heat pumps.
Notes
11.1: Rotational Dynamics
01 Rotational motion $\omega$, $\alpha$, $\theta$ 3.11.1.1 → 02 Angular acceleration equations Rotational SUVAT 3.11.1.2 → 03 Torque $T = Fr$ 3.11.1.3 → 04 Moment of inertia $I = \sum mr^2$ 3.11.1.4 → 05 Newton's second law for rotation $T = I\alpha$ 3.11.1.5 → 06 Angular momentum $L = I\omega$ 3.11.1.6 → 07 Angular impulse $\Delta L = T\Delta t$ 3.11.1.7 → 08 Rotational work & power $W = T\theta$, $P = T\omega$ 3.11.1.8 → 09 Rotational kinetic energy $E_k = \frac{1}{2}I\omega^2$ 3.11.1.9 → 10 Flywheels in machines Flywheel 3.11.1.10 →
11.2: Thermodynamics & Engines
11 First law of thermodynamics $Q = \Delta U + W$ 3.11.2.1 → 12 p-V diagrams Work = area under curve 3.11.2.2 → 13 Thermodynamic processes Isothermal, Adiabatic 3.11.2.3 → 14 The four-stroke petrol engine (Otto cycle) Otto cycle 3.11.2.4 → 15 The diesel engine cycle Diesel cycle 3.11.2.5 → 16 Comparing petrol and diesel engines Compression ratio 3.11.2.6 → 17 Engine power output Indicated, Brake power 3.11.2.7 → 18 Engine efficiency Thermal, Mechanical 3.11.2.8 → 19 Second law of thermodynamics Source, Sink 3.11.2.9 → 20 Heat engines and Carnot efficiency $\eta = 1 - T_C/T_H$ 3.11.2.10 → 21 Limitations of real engines Irreversibilities 3.11.2.11 → 22 Reversed heat engines Refrigerator, Heat pump 3.11.2.12 → 23 Coefficients of performance COP 3.11.2.13 →
Σ Key Equations
On Data Sheet
Not on Data Sheet
Rotational kinetic energy
$$E_k = \frac{1}{2}I\omega^2$$
$I$ = moment of inertia (kg m$^2$), $\omega$ = angular velocity (rad s$^{-1}$).
First law of thermodynamics
$$Q = \Delta U + W$$
$Q$ = heat supplied, $\Delta U$ = change in internal energy, $W$ = work done by the gas.
Torque
$$T = Fr\sin\theta$$
$F$ = force (N), $r$ = distance from axis (m), $\theta$ = angle between $F$ and $r$.
Newton's second law (rotation)
$$T = I\alpha$$
Net torque = moment of inertia $\times$ angular acceleration.
Angular momentum
$$L = I\omega$$
Conserved when no external torque acts. Units: kg m$^2$ s$^{-1}$.
Rotational work and power
$$W = T\theta \qquad P = T\omega$$
Analogous to $W = Fs$ and $P = Fv$ for linear motion.
Carnot efficiency
$$\eta_{\max} = 1 - \frac{T_C}{T_H}$$
$T_C$ = cold sink temperature (K), $T_H$ = hot source temperature (K). Temperatures must be in kelvin.
COP (refrigerator)
$$\text{COP}_{\text{ref}} = \frac{T_C}{T_H - T_C}$$
How effectively a refrigerator removes heat from the cold reservoir.
COP (heat pump)
$$\text{COP}_{\text{hp}} = \frac{T_H}{T_H - T_C}$$
How effectively a heat pump delivers heat to the hot reservoir. Always $> 1$.
Q Retrieval Practice
Q1. A solid disc has moment of inertia $I = \frac{1}{2}MR^2$. If $M = 4.0$ kg and $R = 0.30$ m, calculate the rotational kinetic energy when $\omega = 20$ rad s$^{-1}$.
  • $I = \frac{1}{2} \times 4.0 \times 0.30^2 = 0.18$ kg m$^2$
  • $E_k = \frac{1}{2}I\omega^2 = \frac{1}{2} \times 0.18 \times 20^2 = 36$ J
Q2. State the first law of thermodynamics and define each term.
  • $Q = \Delta U + W$
  • $Q$ = heat energy supplied to the system.
  • $\Delta U$ = change in internal energy of the gas.
  • $W$ = work done by the gas on its surroundings.
  • It is fundamentally an energy conservation statement: energy in = energy stored + energy out.
Q3. An ice skater pulls her arms in during a spin. Explain what happens to her angular velocity and why.
  • Angular momentum $L = I\omega$ is conserved (no external torque).
  • Pulling arms in reduces her moment of inertia $I$.
  • Since $L$ is constant, $\omega$ must increase to compensate.
  • She spins faster.
Q4. A heat engine operates between a source at 600 K and a sink at 300 K. Calculate its maximum theoretical efficiency.
  • $\eta_{\max} = 1 - T_C / T_H = 1 - 300/600 = 0.50$
  • Maximum efficiency = 50%.
  • No real engine operating between these temperatures can exceed this efficiency.
Q5. Explain the difference between an isothermal and an adiabatic process.
  • Isothermal: Temperature stays constant ($\Delta U = 0$), so $Q = W$. The process must be slow enough for heat to flow in or out. On a p-V diagram: gentle curve ($pV = \text{const}$).
  • Adiabatic: No heat transfer ($Q = 0$), so $W = -\Delta U$. Happens when the process is fast or the system is insulated. On a p-V diagram: steeper curve ($pV^\gamma = \text{const}$).